Second-order necessary conditions of the Kuhn-Tucker type in multiobjective programming problems

Control and Cybernetics

vol.

28

(1999) No. 2

Second-order necessary conditions of the Kuhn-Tucker type in multiobjective programming problems

by B. Aghezzaf

Departement de MatMmatiques et d'Informatique, Faculte des Sciences Ain-chock Universite Hassan II

B. P. 5366, Marif, Casablanca, Morocco.

E-mail: Aghezzaf©facsc-achok. ac. ma

Abstract: In this paper, we are concerned with a multiobjec- tive programming problem with inequality constraints. We develop second-order necessary condition of the Kuhn-Tucker type for effi- ciency and prove that the condition holds under a constraint qual- ification. Moreover, we give some conditions which ensure that the constraint qualification holds.

Keywords: Multiobjective programming, efficient solutions, se- cond-order necessary conditions, second-order constraint qualifica- tion

1.

Introduction

In this paper, we deal with second-order necessary conditions for the multiobjec- tive programming problems with inequality constraints. We shall give second- order constraint qualification, and derive the second-order necessary conditions of the Kuhn-Tucker type for a feasible solution to be efficient for the problem.

Moreover, we shall give some constraint qualifications which are also sufficient conditions for the second-order constraint qualification. Our work is in the spirit of Kawasaki (1988) but in the context of multiobjective programming problems.

The paper is organized as follow. In Section 2, we formulate the multiob- jective programming problems with inequality constraints, and provide some definitions and basic results, which are to be used throughout the paper. In Section 3, following Kawasaki, we define two kinds of second-order approxima- tion sets to the feasible region, and using them, we give second-order necessary conditions of the Kuhn-Tucker type for a feasible solution to be efficient for the multiobjective programming problems. In Section 4, we present various kinds of conditions which guarantee the constraint qualification given in Section 3.

2. Preliminaries

In this section we shall introduce some notations and definitions, which are used throughout the paper. Let IRn be the n-dimensional Euclidean space, and let x = (x1, ... , xnf and y = (y_1, ... , Ynf be in IR.n. We shall denote the inner product of x and y by x · y

= I:Z=l

^XiYi.

For any two vectors x and y in IRn, we shall use the following conventions:

x-:;_y iff Xi-:;_ Yi, i = 1, ... ,n, x::;_y iff x-:;_y and X =J y, x<y iff Xi

<

Yi, i = l, ... ,n.

For any two vectors x and y in JR², we sha.ll use the following conventions:

x

<

y iff x1

<

Y1 or

=lex

x <zex y iff X1

<

^Yl or

XJ = Y1 and xz-:;_ yz, x1

=

Yl and xz

<

yz. the subscript lex is an abbreviation for lexicographic order.

Let f : IRn ---+ IR¹ and g : IRn ---+ IRm. We assume that the functions are twice continuously differentiable on IRn, and for any vector y, we denote the Jacobian (resp. the Hessian) off and gat x E IRn by \lf(:r) and \lg(x) (resp. \7² f(x) (y, y) and \7² g(:r) (y, y)).

We consider the following multiobjective optimization problem:

(P)

^mm _{s.t x} ^f(x), _{EA= {x E IRnlg(x) ~ 0},}

Due to the conflicting nature of the objectives, an optimal solution that simultaneously minimizes all the objectives is usually not obtainable. Thus, for problem (P), the solution is defined in terms of an efficient solution, Yu (1985).

DEFINITION 2 .l A point x E A is said to be an efficient solution to problem (P), if there is no x EA such that f(x) ::;_ f(x).

Let

x

^E A be any feasible solution to problem (P), and let E be the subset of indices defined by

E = {j E {1, 2, ... , m} I gj(x)

=

0 }. (1) DEFINITION 2.2 The tangent cone to A at x EA is the set defined by

T1(A,x)=

{ y E JRn l3xn E A, 3tn ---+

o+

such that :rn

=

x

+

tny

+

o( tn) } , (2) where o(tn) is a vector satisfying llo(tn)ll/tn---+ 0.

Second-order necessary conditions of the Kuhn- Tucker type

For each i = 1, 2, ... , l, we shall define the nonempty sets Qi and Q by Qi

=

{x E IRnlg(x) ~ 0, fk(x) ~ fk(x), k

=

1, 2, ... , l, and k

#

i}, Q

=

^{{x E IRnlg(x) ~} 0, f(x) ~ f(x)}.

In the case l = 1, we set Qi = A.

DEFINITION 2.3 The linearizing cone to Q at x is the set defined by

3. Second-order necessary conditions

215

Following Kawasa.ki (1988), we shall define two kinds of second-order approx- imation sets to the feasible region. They can be considered as extensions of T1 (A, x) and K₁ respectively. Using them, we shall give the second-order con- straint qualification, under which we derive second-order necessary conditions of the Kuhn-Tucker type for a. feasible solution

x

E A to be an efficient solution to problem (P).

DEFINITION 3.1 The second-order tangent set to A ^{at x E} A is the set defined by

{ (y' z) E JR²n I :lxn E A, :ltn ---+

o+

such that

- 12 (2)}

X+

tny

+

2tnz

+

0 tn ,

where o(t;) is a vector satisfying llo(t;)ll/t;---+ 0.

They-section of T2(A, x) is defined by T2(A, x)(y)

=

{z E

mn

^{I (y, z) E} T2(A, x) },

DEFINITION 3.2 The second-order linearizing set to Q at x is the set defined by

K2

=

^{{(y, z) E JR2n}

I

(\7 fi(x)y, \7 fi(x)z

+

\72 _fi(x)(y, y))T ~ (0, Of, i = 1, 2, ... , l,

lex

(\lgJ(x)y, \lgJ(x)z

+

^\72gj(x)(y,

y)f

~ (0,

of,

j E E }.

lex

They-sections of K₂ is defined by

It is obvious that K₂(y) is closed convex set for each direction yE lR".

Before deriving second-order necessary conditions for efficiency, we shall give the following lemma, which shows the relationship between the second-order tangent sets T2 ( Qi, x) and the second-order linearizing set K_{2 .}

LEMMA 3.1 Let x E A be any feasible solution to problem (P ). Then, we have

n

l co[T2(Qi,x)(y)] c;;; K2(y)

i=l

for arbitrary direction y of JRn, where co[T2 ( Qi, x) (y)] denotes the closed convex hull ofT2(Qi,x)(y).

Proof. First, we shall show that T2(Qi,x)cK?,, i=l,2, ... ,1, where

K?,

=

^{{(y,z) E JR2n}

^I

(V fi ( x)y, V fi ( x)z + V2 fi ( x) (y, y)) r ;;;; (0, of, i

=

1, 2, ... , l, k

=1

i

lex

(Vgj(x)y, Vgj(x)z

+

V 2gj(x)(y, y)f;;;; (0, O)T, j E E }.

lex

For any fixed i

=

1, 2, ... , l, let (y, z) be any element of T2(Qi, x). Then, there exist xn ^E Qi, tn ---+

o+

such that

xn

=

x + tny + 1j2t;z + o(t;).

I

By Taylor's expansion,

fk(xn) ⁼ fk(x) +tnVfk(x)y

+1/2t~(V fk(x)z + V2 fk(x)(y, y)) + o(t~), k

=

1, 2, ... , l, k

=1

i

=* 0;:::: tn V !k(x)y

- +1/2t~(Vfk(x)z+V²fk(x)(y,y))+o(t~), k=1,2, ... ,l,k=li.

Thus, for n large enough, V fk(x)y;;;; 0 and if V fk(x)y = 0, we obtain

which after letting n ---+ +oo, gives

which implies

(V !k(x)y, V !k(x)z + V2 !k(x)(y, y)f ;;;; (0, of, k

=

1, 2, ... , l, k

=I

i.

lex

Similarly, we have

(Vgj(x)y, Vgj(:'f)z

+

V 2gj(x)(y, y))r;;;; (0, o)r, j E E.

lex

Second-order necessary conditions of the Kuhn-Tucker type

Hence,

T2 ( Qi, x)

c

K~, i = 1, 2, ... , l,

and, for any arbitrary direction y of JRn, we have T2(Qi,x)(y)cK2(y), i=l,2, ... ,l.

Since, KHy) is a closed convex set and i is arbitrary, we have

l l

n

co[T2(Qi, x)(y)J <:;;;

n

^{K~(y) <:;;; K2(y)}

i=l i=l

217

In Lemma 3.1, the converse inclusion does not hold, in general. Therefore,

•

it is reasonable that we assume that

l

K2(y) <:;;;

n

co[T2(Qi,x)(y)] (4)

i=l

holds for the direction y in order to derive the second-order necessary condi- tions for a feasible solution

x

^E A to be an efficient solution to problem (P).

The condition ( 4) is called the second-order generalized Guignard constraint qualification for the direction y, and we will refer ( 4) as to the second-order (GGCQ). Since K2(0)

=

K1 and T2(A, x)(O)

=

T1 (A, x), the above condition contains (GGCQ) of Maeda (1994).

First-order sufficient condition for efficiency is that the following system have no nonzero solution y:

\lf(x)y ~

o,

\lgE(x)y ~

o,

⁽⁵⁾

and the condition of Kuhn-Tucker type for efficiency is equivalent, Ma.eda (1994), Marusciac (1982), Singh (1987), to the inconsistency of the following system :

v

f(x)y :::;

o,

\lgE(x)y~O,

The gap between (5) and (6) is caused by the following directions:

V f(x)y =

o,

\lgE(x)y ~

o.

A direction y which satisfies (7) is called a critical direction.

For the sake of simplicity, we will use the following notation:

Fi(y, z) =(V fi(x)y, \l J;(x)z

+

\7² fi(x)(y, y))r, G1(y, z)

=

(\lg1(x)y, \lg1(x)z

+

\l2g1(x)(y, y))r.

(6)

(7)

Now, we are in a position to state the primal form of our second-order necessary conditions.

THEOREM 3.1 Let

x

^{EA be an efficient} solution to problem (P). Assume that the second-order (GGCQ) holds for any critical direction. Then, the follow·ing system has no solution (y, z):

i=1,2, ... ,l, for at least one i,

't:/j ^E E.

(8)

Proof. Suppose on the contrary that there exists (y, z) such that (8) hold.

Thus, we have z E K₂(y). Without loss of generality, we may assume that F1(y,z) <lex 0,

Fi(y,z)~ 0, i=2, ... ,l,

- lex

by assumption, we have z E co[T2(Ql,x)(y)].

Hence, there exists a sequence {zn} of co[T2(Q¹,x)(y)] converging to z. Each zm can be written as a convex combination of some elements of T2(Q\x)(y), say

zmi, ...

,zms. For each m = 1, 2, ... and k = ] , 2, ... , s, since zmk E T₂(Q¹, x)(y), by definition, there exist x~k ^E Q¹ and t~k ---> +0 such that

x~k

= x +

t~ky

+

~(t~k)2zmk

+

o((t~k)2).

Then, for all n, we have

fi(x);:::. !i(x~k) = fi(x)

+

t~k\lfi(x)y

- +~(t~k)2(\l !i(x)zmk + \72 fi(x)(y, y)) + o((t~k)2),

O;::::g1^(x~k)

=

g₁(x) +t~k\lg1^(x)y

- +~(t~k)2(\lgj(x)zmk

+

\l2g1(x)(y, y))

+

o((t~k)2).

Then, we have

Fi(y,z)~ 0, for i=2,3, ... ,l,

-lex

G₁(y, z)

:s;

0, for j E E.

-lex

(9)

On the other hand, since

x

EA is an efficient solution to problem (P), for all n, we have

h(x~k);:::.hx)

h (x)

:s;

h(x~k) = h (x)

+

t~k\7 h (x)y

- +~(t~k)2(\l h(x)zmk + \72 h(x)(y, y)) + o((t~k)2).

Then, we have

0 ~ F1 (y, zmk). (10)

lex

Second-order necessary conditions of the Kuhn- Tucker type 219

From (9)-(10), it follows that

From the linearity and the continuity of Fi and G₁ with respect to z, we have F1(y,z)?_ 0,

-lex

Fi(y,z)-:5_ 0, i=2,3, ... ,Z,

-lex

G₁(y,z) ::;_ 0, j E E.

-lex

This is a contradiction.

In particular, the system (8) has no solution of the form (O,z).

•

From the above theorem we get the first-order necessary conditions for effi- ciency which were already given in Maeda (1994).

Now, we shall state the dual form of Theorem 3.1

THEOREM 3.2 Let x satisfy the assumptions of Theorem 3.1. Then, for each critical direction y, there exist multipliers A E JR.z and f-L E !Rm such that

i=l j=m

2:::::>-i'Vfi(x) + L

^tt1'Vg₁(x)

= o,

i=l j=l

(~

^{Ai \72 fi(x)}

⁺

¹

~

^{f-LJ \72g1}(x)+) (y, y)

~ ^o,

A> 0, f-L~O, /-Lj = 0 '<:/j rf_ E(y),

E(y) = { j E {1, ... 'm} I gj(x) = 0, 'Vgj(x)y = 0 }.

Proof. Let y be a critical direction. Then, the system:

\7 f(x)z

+

\7² f(x)(y, y) ~ 0, 'Vge(y)(x)z

+

\7²ge(y)(x)(y, y) ~ 0,

has no solution z, which is equivalent to

\7 f(x)z

+

\7² f(x)(y, y)~ ::;

o,

'Vge(y)(x)z

+

\7²ge(y)(x)(y, y)~ ::; 0,

-~ < 0,

(11)

(12)

having no solution z E !Rn, ~ ER. By Slater's theorem of the alternative, Mangasarian (1969), there exist multipliers A and fl such that either (13) or

(14) holds:

i=l j=m

:Z:::.>'i

'V fi(x) +

L

P,j 'V 9.i(x)

=

o,

i=l j=l

(~A,

^{\72 f,(x)}

+';

^{MJ \72}

^g,(x))

^{(y, y)}

^>

^0,

A~ 0, p,~O, P,j

=

0 'l!j tf_ E(y);

i=l j=m

LAi'Vfi(x)

+

L P,j'Vgj(x)

= o,

i=l j=l

(~

^A;\72 [;(X)+%' MJ \7²

g,(X))

^{(y, y)}

^>

^0,

A> 0, p,~O, p,₁

=

0 'l!j tf_ E(y).

(13)

(14)

Let us assume that (14) does not hold. This is equivalent to the inconsistency of the system

i=l j=m

LAi'Vfi(x)

+

L P,j'Vgj(x)

= o,

i=l j=l

(~

^{Ai \72 fi(x)}

⁺ j~

^{P,j 'V2}gj(x)) (y, y)-s.1

=

0, A

>

0, s ~ 0, p, ~ 0, P,j

=

0 'l!j tf_ E(y).

By Tucker's theorem of the alternative, Mangasarian (1969), there exist z and

t;::::

0 satisfying

'V f(x)z

+

\7² f(x)(y, y)t:::; 0, 'VgE(y)(x)z

+

'V²9E(y)(x)(y, y)t :s; 0.

Since (12) has no solution, we have t

=

0; hence, 'V f(x)z:::; 0, 'V 9E(y)(x)z :s; 0.

On the other hand, 'V f(x)y

=

^o,

'VgE(y)(x)y =

o,

because y is critical. Thus, 'Vf(x)(y+EZ):::; o, 'VgE(x)(y

+

Ez) ~ 0,

'VgE\E(y)(x)y < 0,

for any sufficiently small E > 0, which contradicts the first-order necessary conditions for efficiency. This completes the proof. •

Second-order necessary conditions of the Kuhn- Tucker type

ExAMPLE 3.1 Consider the following problem:

min (h(xl,x2),h(xl,x2)) = (xf - x~,x2- xl), s.t gl(x1,x2)=x1+x2~0,

92(XJ,X2) = X1 - X2 ~0.

The x = (x1,x2)T = (O,Of is an efficient solution and Q1 = {(xl, X2)T E JR2Ix1 = X2, X1 ~ 0},

Q2 = {(xl, x2f E JR²Ixt =X~, XJ ~ 0}.

Hence, we have

K2(0) = co[T2(Q 1,x)(O)]

n

co[T2(Q2,x)(O)]

= {(z1, z2f E IR2Iz1 = z2, ZJ ~ 0}.

Moreover, for each critical direction y -=J 0, we have K2(y) = co[T2(Q1, x)(y)]

n

co[T2(Q2, x)(y)]

= {(z1,z2f E IR2Iz1 = z2}.

221

Therefore, the {GGCQ) holds at x for any critical direction y. Then, for (.\J,A2,f.-Ll,f.l-2) = (1,1,0,1), we have

t , Ai \7

h

(X) +

t,

^{p, j \7 g j ( x) =}

^G)

^{+ (}

^~

^{1) + (}

^~1)

⁼

^G),

and for each critical direction y,

(yl,Y2)(t,,\i\72

.fi(x) + t,p,j\72

gj(x))(yl,Y2f =

y~ -y~ ⁼

^0.

4. Sufficient conditions for the second-order generalized Guignard CQ

In the preceding section, we introduced the second-order constraint qualification (4). In this section, we shall present several conditions which guarantee (4).

THEOREM 4.1 Let y be any critical direction. If any of conditions (a) through (e) holds, then (4) holds.

(a) Ben-Tal's Constraint Qualification (BTCQ): For each i

=

1, 2, ... , l, the system

\7 .fk(x)v + \7 2 .fk(x)(y, y)

<

0, k = 1, 2, ... , l and k -=J i,

\lgj(x)v

+

^\72gj(x)(y,y)

<

0, j E E(y), has a sohltion v E IRn.

(b) Cattle-Type Constraint Qualification (CCQ): For each i

=

1,2, ... ,1, the system

\lfk(x)v

<

0, k

=

1,2, ... ,l and k -=J i,

\lgj(x)v < 0, j E E, has a solution v E IRn.

(c) Slater's Constraint Qualification (SCQ): h i = 1,2, ... ,1, and ,9j, j 1, ... , m are all convex on JR.n, and for each i

=

1, 2, ... , l, the system

!k(x)

<

fk(x), k

=

1,2, ... ,l and k # i, g₁(x) <0, j=1, ... ,m.

has a solution x E JR.n.

(d) Mangasarian-Fromovitz's Constraint Qualification (MFCQ): \lj;(x), ~

=

1, 2, ... , l are linearly independent and the system

\lj;(x)v=O, i=l,2, ... ,1,

\lgJ(x)v

<

0, j E E.

has a solution v E JR.n.

(e) Linear Constraint Qualification (LCQ): j;, i

=

1,2, ... ,1, and ,9j, j E E, are all linear.

Proof. To show that if any of conditions (a) through (e) holds then (4) holds, it is only sufficient to show that (a) guarantees (4); since (d) yields (b) which in turn yields (a.), and (c) yields (b).

Next, we show that (a.) guara.ntees(4). Let z be any element of K2(y). Then, we have

\1 j;(x)z

+

\7² j;(x)(y, y):;, 0, i = 1, ... , l,

\lgJ(x)z

+

\1²gJ(x)(y, y) ~ 0, \/j E E(y).

First, we shall show that, for each i

=

1, 2, ... , l, z E T2(Qi,x)(y).

By assumption, there exists V E

mn

such that

\1 fk(x)v

+

\7² fk(x)(y, y)

<

0, k

=

1, 2, ... , l and k # i,

\lgJ(x)v

+

\1²gJ(x)(y,y)

<

0, j E E(y),

For any positive sequence { tn} converging to 0, we shall define the sequence { zn} converging to z by

zn = z

+

tn(v-z).

From (15), for each n, we have

\lfk(x)zn+\1²fk(x)(y,y)<O, i=1, ... ,l a.nd k#i,

\lgJ(x)zn

+

\1²gJ(x)(y,y)

<

0, j E E(y).

For each zn, n

=

1, 2, .... , and a.ny positive sequence {J.Ls} converging to 0, we shall define the sequence { xns} converging to x by

Xns =X+ J.lsY

+

1/2J.L~Zn

+

o(J.L~).

Then, for all s sufficiently large, we have Xns =X+ J.lsY

+

1/2J-i~Zn

+

o(J-i~), fk(xns)

=

fk(x

+

J.lsY

+

1/2J.L~Zn

+

o(J.L;))

=

!k

(x)

+

J.ls \1

!k

(x)y

+ l/2J-i;

('V

!k

(x)zn

+

\7²

!k (

x) (y, y))

+

o(J.L;)

~fk(x), k=1,2, ... ,l and k#i,

Second-order necessary conditions of the Kuhn- Tucker type 223

and

gj(Xns) = gj(X

+

{lsY

+

1j2{l;Zn

+

o({L;))

=

gj(x)

+

^{!8 '\lgj(x)y

+

l/2{t;('\lgi(x)zn

+

^'\72g₁(x)(y, y))

+

o({L;)

<

gj(x)

=

0, j E E.

For j rf_ E, from the continuity of gj, it follows that gj(xns)

<

0, all s sufficiently large.

Hence, we have that

xns E Qi, all s sufficiently large.

and

Since T_{2 (} Qi, x) (y) is closed, we have z E T2(Qi,x)(y), i

=

1,2, ... ,l.

Therefore, we have

It is easily proved that the condition (e) quarantees (4). This completes the

~~ D

Acknowledgement

The author would like to thank the anonymous referees for their many valuable comments and helpful suggestions.

References

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